Original article Power for mapping quantitative trait loci in crosses between outbred lines in pigs Luis Gomez-Raya* Erling Sehested Department of Animal Science, Agricultural University of Norway, Box 5025, N-1432 As, Norway (Received 28 May 1998; accepted 12 May 1999) mapping quantitative trait loci using crosses between segregating populations was studied in pigs Crossing generates gametic disequilibrium and increases heterozygosity The condition for a heterozygosity among F individuals to l be greater than in either line at crossing is that allele frequency should be lower than 1/2 in one line and higher than 1/2 in the other line Maximum expected power and expected risk were used to compare hierarchical backcross (each boar mated to several Abstract - Power for boar mated to several sows; sows; contrast within boars), hierarchical intercross contrast within both boars and sows) and traditional intercross (each boar mated to one sow; contrast within both boars and sows) The use of hierarchical designs (each (back- increased power for traits with low or intermediate heritabilities QTL effects and low heritabilities the hierarchical backcross design gave the highest expected power but also the highest risk There is not a general design which allocates resources in an optimum fashion across situations (heritabilities, QTL effect, heterozygosity) A compromise between designs with high power and low risk is suggested A hierarchical backcross design of 400 piglets and heterozygosity 0.68 requires between four (maximum expected power) and eight boars (minimum risk) to detect a QTL of 0.5 phenotypic standard deviations Selection of extreme individuals in parental lines increased power up to 21 % Commercial crosses are proposed as an alternative to experiments for QTL mapping @ Inra/Elsevier, Paris and For small cross gene intercross) mapping / quantitative trait locus / statistical power / hierarchical design / pig Résumé - Puissance de détection des loci effets quantitatifs dans les croisements lignées non consanguines chez le porc On a étudié chez le porc la puissance de détection des loci effets quantitatifs partir de croisements entre populations en I ségrégation Le niveau d’hétérozygotie chez les individus F est supérieur celui de entre * Correspondence and reprints luis.gomez.raya!akvaforsk.nlh.no E-mail: l’une ou l’autre des lignées en croisement si les fréquences alléliques sont supérieures 1/2 dans l’une des lignées et inférieures 1/2 dans l’autre lignée La puissance maximale espérée et le risque espéré ont été considérés pour comparer les croisements en retour hiérarchiques (chaque verrat est accouplé plusieurs truies : contrastes intra-verrat), l’intercroisement hiérarchique (chaque verrat est accouplé une truie ; contraste intra-verrat et intra-truie) L’utilisation de ces deux types de schémas hiérarchiques a augmenté la puissance de détection pour les caractères héritabilités faibles ou intermédiaires Pour les CaTLs petit effet et les héritabilités faibles, le schéma hiérarchique avec croisement en retour a donné la puissance espérée maximale et aussi le risque le plus élevé Ce n’est pas le mờme schộma expộrimental qui alloue les ressources dune faỗon optimale quand on considère des situations différentes pour l’héritabilité, les effets de QTL et le niveau d’hétérozygotie Un schéma hiérarchique avec croisement en retour de 400 porcelets et une hétérozygotie de 0,68 requiert entre quatre verrats (puissance espérée maximum) et huit verrats (risque minimal) pour détecter un QTL avec un effet de 0,5 écart-type phénotypique La sélection d’individus extrêmes dans les lignées parentales a augmenté la puissance jusqu’à 21 % Les croisements entre lignées commerciales sont proposés comme alternatives aux expérimentations de détection de QTL © Inra/Elsevier, Paris détection de gènes / hiérarchique / porc locus effet quantitatif / puissance statistique/ schéma INTRODUCTION Crosses between inbred lines and analyses within outbred lines have been proposed for quantitative trait loci (QTL) detection In the first approach, inbred lines are assumed to be fully homozygous for alternative alleles at both marker and QTL Consequently, F individuals are fully heterozygous l and gametic disequilibrium is maximum High heterozygosity allows high segregation and, therefore, high power With maximum gametic disequilibrium, a specific allele at the marker is associated with a specific allele at the QTL in all families Consequently, increased power is achieved because a lower number of contrasts is needed using across family analyses based on a larger number of observations per contrast In the second approach, analyses within outbred lines, both marker and QTL are segregating Power in crosses between inbred lines is much higher than in analyses within outbred lines A combination of both approaches has assumed that the lines are segregating at the marker but fixed for alternative alleles at the QTL [1, 2! Another alternative, not explored yet, is to consider parental lines with different allele frequencies at the QTL because of different selection history Conventional types of crosses between inbred lines are intercross and backPower of an intercross versus a backcross design is increased because the number of segregating rneioses in an intercross is twice as many as in a backcross Experiments conducted in pigs have restrictions in family structure to a maximum of approximately ten piglets per litter A small number of full-sibs may result in low power !9! A hierarchical structure, such as a few boars mated to several sows each, allows larger subgroups of progeny inheriting alternative alleles from each boar, which may increase power It has been proposed to use selective genotyping to increase power for QTL detection [10, 11] Selective genotyping is efficient when the cost of cross progeny is less than the cost of genotyping Another use of selection, yet, would be to use extreme individuals for the quantitative trait as parents to produce the F This approach increases the frequency of l l heterozygotes in the F and, therefore, statistical power The objectives of this paper are: 1) to investigate the degree of disequilibrium generated when crossing two populations segregating at two loci; 2) to investigate under which conditions the heterozygosity (and consequently power) in the cross of two lines segregating at a QTL is higher than within either line; 3) to investigate power using hierarchical designs; 4) to investigate optimum allocation of resources for a given experiment size; and 5) to investigate the effect of selection of extreme individuals in the parental lines on heterozygosity and power growing not considered THEORY 2.1 of Underlying genetic model and assumptions The purpose of the experimental crossing is the mapping of QTL by the use genetic markers such as microsatellites It was assumed that recombination events did not occur between marker and QTL and that all offspring were informative for the marker These simplifying assumptions were made to reduce the number of parameters to be shown in the results The impact of these assumptions in practical scenarios is addressed in the discussion An underlying mixed inheritance additive model was assumed with an additive biallelic locus segregating at frequencies of the favourable allele p B and pc for lines B and C, respectively The QTL was assumed to be in HardyWeinberg equilibrium and in gametic phase equilibrium with the polygene in each parental line The variance attributable to the QTL in units of the residual phenotypic variance in linei was: (1 i 2p Pi)a2 /(u! + , )a E where p is the frequency of the favourable allele in line i, cr is the QTL i effect, and (a A + 0is the residual phenotypic variance with and OF2 E ,2 E ,2 A being the additive variance attributable to the polygene component and the environmental variance, respectively The heritability comprising QTL and polygene variation in parental linei was: vb(i) = where h!is the residual heritability attributable to the polygene component, and has value a A /(U + 0,2 E ) Parameters h!and are usually not known A and only estimates of h are available for some traits Different heritabilities (i) in the two parental populations were not assumed because of the magnitude of the sampling variance of the estimates of heritability and because parental populations may have been raised under different environmental conditions For simplicity, power computation in this paper was carried out for a given QTL effect and constant residual heritabilities among F individuals Therel fore, results are valid for a variety of situations with respect to the parental populations segregating at a QTL, not only at a different frequency, but also 2Q V with sible different heritability (comprising QTL and polygene) The highest posheritability (QTL and polygene) in the parental populations (denoted as l h (max)) leading to a given heterozygosity at the QTL among F individuals was computed as an indication of the values of heritability in practical situaa tions The impact of this assumption on power is addressed in the Discussion Three levels of heterozygosity in the F were considered: 1, 0.68 and 0.32 A l heterozygosity of occurs when the QTL is fixed for alternative alleles A heterozygosity of 0.68 may occur for a range of situations in parental populations = B (e.g PB 0.8; pc 0.2 and p 0.68; pc 0) A heterozygosity of 0.32 represents the situation where the two lines segregate at the same frequency B (e.g p 0.8; pc 0.8) or when one allele is absent from one population and segregating at a frequency lower than 0.5 in the other population (e.g B p 0.32; p 0.0) The advantages of crossing experiments aimed at QTL c = mapping are: 1) linkage disequilibrium between QTL and marker alleles in the ; l F and 2) increased heterozygosity among F individuals l = = = = = = 2.2 Gametic disequilibrium in the cross of two outbred lines If the two populations are fixed for alternative alleles at both marker and l QTL, then gametic disequilibrium is maximum in the F Therefore, power is increased with respect to within-family analysis If the two parental populations are segregating at the same frequency, then the resulting F population is in l linkage equilibrium and no benefits are expected from crossing If the genes are segregating at different frequencies in the parental populations then some degree of disequilibrium will be generated For a more general description of this problem, gametic disequilibrium will be computed at two loci, M and N, that can be either two markers or one marker and one QTL Gametic disequilibrium between markers can be estimated if estimates of allele frequencies in the parental lines at crossing are available Let f and fb,! be vectors of allele 6M frequencies at locus M (with k alleles) and at locus N (withg alleles) in eM parental population B Similarly, f and f!,V are the corresponding vectors of allele frequencies at loci M and N in parental line C Assuming that the two loci are in linkage equilibrium within the line then the matrix with gametic frequencies at each pair of alleles for line B is: The matrix of gametic frequencies in line C is F eMjV f f!!, The matrix ’M with gametic frequencies for each pair of alleles at loci M and N in the F of l = the cross between lines B and C is given by: F 1/2(F + F N beM bMN EMN = )1 The allele frequencies for loci M and N in the F cross are fbcm ! F bcMN l 1’ F respectively In these equations represents the unit and f6!N , beMN vector The gametic disequilibrium matrix is then obtained simply by D MN MN FB,MN - (fb,M f6!NO Elements of matrix D are the disequilibrium values for all possible combinations of alleles at loci M and N Define the disequilibrium parameter between two loci M and N, S2 as the sum of absolute values of , MN all elements of matrix D computed by 52,!,Ir, 1’ abs(D where ’abs’ MN )1, MN denotes the absolute value at each element of the matrix between brackets MN Therefore, S2 measures the general degree of association between alleles at loci M and N The value of 52,,,1,! ranges between and For example, Q MN is when lines at crossing are fixed for alternative alleles and disequilibrium is maximum and Q is when the lines at crossing are segregating at the same MN frequency for each allele and disequilibrium is null = = = 2.3 Heterozygosity in the cross of two outbred lines Consider two outbred lines B and C segregating at a biallelic QTL with alleles Q and q Assuming random mating, the frequency of heterozygous individuals at the QTL among the F individuals is: l It is shown in Appendix that the necessary condition for increased with respect to either line at crossing is that one line is a frequency of the favourable allele greater than 1/2 and the other line at an allele frequency lower than 1/2 heterozygosity segregating at 2.4 QTL mapping designs for the cross between two outbred lines Power computation was carried out assuming that allele frequencies at the marker were not very different in the two parental populations, and consequently, gametic disequilibrium was small and ignored However, the increased heterozygosity in the F boars can be used to study segregation, within families, of a QTL associated to a marker by recording performance and inheritance of alternative marker alleles in the next generation (backcross or intercross depending on the mating of F boars to one of the parental l lines or F sows, respectively) Therefore, the approach that can be followed l is the same as in QTL mapping within outbred lines but with an increased l heterozygosity among F individuals Three alternative designs are considered in this section: hierarchical backcross design, traditional intercross design and hierarchical intercross design Contrasts in hierarchical designs can use large subgroups of progeny of a mixture of half- and full-sibs within boar 2.4.1 Hierarchical backcross design In a hierarchical backcross design, b boars from the F are mated to s sows l each to produce p piglets per litter Sows can be from any parental line A statistical model to analyse the data could be: where y2!kc is the lth observation of phenotype on a piglet with marker allele k inherited from boari when mated to sow j, bo is the fixed effect of boar i, so2! i is the fixed effect of sow jmated to boar i, m is the fixed effect of marker ijk allele k inherited from boar i, and e is the residual random error Model ijkl (2) is a three level hierarchical design in which marker alleles are nested within sows, which in turn are nested within boar For simplicity, sows and boars were assumed fixed It is not strictly correct because the model does not account for relationships between animals Power for model (2) was computed following the x approach of Gelder2 mann [5] and Weller et al [14] comparing the square of the difference between the two progeny subgroups inheriting alternative alleles from their boar (SDP) for each boar family to the expected squared difference under the null hypothesis Therefore, the alternative hypothesis is a QTL linked to the marker and the null hypothesis is the absence of a QTL linked to the marker For a more detailed description of the method and discussion of the assumptions, see Weller et al !14! Briefly, the assumptions are complete linkage between a fully informative marker and a QTL, and the analyses are carried out within families For simplicity, it is assumed that there is no common environmental variance and litter size is fixed at 10 The statistical tests to reject or to accept the null hypothesis are based on the distribution of the SDP The summed SDP values divided by the squared standard error of the contrast (SE follow a central k distribution ) with b degrees of freedom under the null hypothesis for a large sample size or when phenotypic variance is known For the alternative hypothesis (nc, x b), where nc is the non-centrality parameter of a noncentral x2distribution with b degrees of freedom The value of the nonwhere a + 6(l - p BC b he centrality parameter is nc ) B and c! + 6(1 - 2p for backcrosses with lines B and C, respectively ), c (Appendix 2), a is half the difference between the two alternative homozygotes, is the dominance deviation, and SE! = [(1 + (1/4)shr - l/2h!)/(sp)] N ) E(SDP/SE = 0!/SE!, = = [equation (A3) in Appendix 3! Following Weller et al [14], the power to detect a segregating QTL was ! p[x b) < T], where is the probability of computed as - ! (nc, a type error, [p(x b) < committing (nc, 2 T] is the probability of a x value under the alternative hypothesis (non-central Xwith parameter nc and b degrees of freedom) less than T, with T being the value of the central x (b) for a given significance level of committing a type error = 2.4.2 Traditional intercross design In the intercross design, inbred lines B and C are crossed to produce the that are intercrossed among themselves In the traditional intercross design b boars are mated to one sow each A linear model allowing testing for the marker-(aTL effects is: l F boars and sows where Y is the hth observation on piglet k with marker genotype jinherited k ij in family i, f is the fixed effect of family i, m is the fixed effect of markerz ij QTL genotype jinherited in family i ( j to 3; homozygous for either allele = heterozygous) and e is the residual error Model (3) would be the one to ijk in the analysis of real data However, to take advantage of the X approach of Geldermann [5] and Weller et al [14] for power calculation, the following linear model can be used: and use where !z!!l is the lth phenotypic observation of piglet inheriting marker allele k from the sow and allele j from the boar, mb is the fixed effect of marker allele ij k i j inherited from the boar, m is the fixed effect of marker allele k inherited from the sow, and ei!xl is the residual error Model (4) is linearly equivalent to model (3) under the assumption of a fully additive underlying genetic model and no sexual imprinting The distribution is approximately x b) and under the alternative hypothesis of (nc, z (nc, X s) for boars and sows, respectively Assuming that the hypothesis being tested is the same either within boars or within sows, the distribution of the for both boars and sows follows a x b+s) under the sum of (2nc, z alternative hypothesis The sum of variables having non-central x distributions z jointly independent also follows a non-central X2distribution with degrees of freedom equal to the sum of the degrees of freedom of the former non-central X2 and non-centrality parameter equal to the sum of the non-centrality parameters of the former variables having non-central x The non-centrality parameter of z the x has a value 2nc BC -C ), z , (b + s) he (p /SE where0 = a + 6(l - PB p and SE!= !(1 - (l/2)!)/p] [equation (A4) in Appendix 3] Similarly to a hierarchical backcross design, power to detect a segregating QTL was computed - p[x2(2nc, 2b) < T], where /3 is the probability of committing as - (3 a type error, !p(2(2nc, 2b) < T] is the probability ofax value under the alternative hypothesis (non-central X2with parameter 2nc and 2b degrees of freedom) less than T, with T being the value of the central X2(2b) for a given significance level of committing a type error ) E(SDP/SE ) ’¿,(SDP /SE = = 2.4.3 Hierarchical intercross design In a hierarchical intercross design, b boars from the F are mated to s sows l each from the F to produce p piglets per litter Power calculation can be l carried out using the linear model: where Y is the mth observation of phenotype on a piglet with marker klm ij allele k inherited from boar i mated to sow j, bo is the fixed effect of boar i ik i, so is the fixed effect of sow j mated to boar i, mb is the fixed effect ij of marker allele k inherited from boar i, ms is the fixed effect of marker ijl allele I inherited from sow ij and e is the residual error Power calculation ijklm using model (5) requires different non-centrality parameters within boars and where sows The non-centrality parameter in boars is: nc b with SE= 4[(1 + (1/4)ph; -1/2h;)/(sp)] c b( ) P B = a + - (Appendix 2), I - p are as for traditional intercross Z for s > (Appendix 3) Values of SEfor s = = che B b e /SE2, design = The non-centrality parameter in sows is nc, s = , /SE B BC bshe where ) c -p PB a+6(I- (Appendix 2), and SE= 4[(1 - ( /2)h$) /p] (Appendix 3) ) E(SDP/SE for both the distribution of the sum of boars and sows follows a x b(1+s)) under the alternative hypothesis +nc.&dquo; b (nc Under the null hypothesis, the distribution of the sum of E(SDP/SE for both ) boars and sows follows a central X + s)) Power to detect a segregating (b( , s QTL was computed as - (3 ! p!x2(ncb + nc b(1 + s)) < T], where /3 is the probability of committing a type error, (p(x + nc,, b(1 + s)) < T] is h (nc the probability of a x value under the alternative hypothesis (non-central x 2 with parameter nci, + nc and b(1 + s) degrees of freedom) less than T, with s T being the value of the central x (b(1 + s)) for a given significance level of committing a type error Assuming independence, = 2.5 Design of experiments using crosses between outbred lines Most experimental costs are in raising, genotyping and recording of a given number of slaughter pigs in the F or backcross Therefore, the parameters to be chosen by the researcher are the number of boars, the number of sows and the number of piglets per sow In practice, it is convenient for the handling of the experiment to fix the number of piglets per sow It is also economical since the use of few piglets per sow would increase the cost in raising a large number of sows It will be assumed from now on that the number of piglets per litter in the experiment is fixed at ten and the question is how to make optimal use of different numbers of boars and sows in the F for a given experiment size l (total number of slaughter pigs) Parameters of interest are the expected power (EP) and its standard deviation (SD): where pr(i) is the probability according to the binomial distribution of having i heterozygous boars [assuming a frequency of heterozygous boars given by i equation (1)], and P is the power with i boars computed according to the previous sections Expected power is utilized for each design to account for random sampling of the boars SD can be used as a measurement of the variation in power Comparison of alternative crossing designs can be done by comparing expected power and its standard deviation at a fixed experiment size The approach used in this paper for optimum allocation of resources is based on the repeated computation of expected power for all possible combinations of the number of boars and sows at a given experiment size assuming constant litter size of ten piglets The power of the mating structure having the highest expected power will be called maximum expected power (MEP) The functions CINV and PROBCHI of SAS [12] were used to compute central and non-central xprobabilities, respectively Another parameter of interest in planning experiments is expected risk which we sampling define among as l F (ER) the probability of having power lower than 0.5 due to individuals in the segregating population: 2.6 Selection in parental lines to increase power for QTL mapping Selection of high ranking individuals in one parental line and of low ranking individuals in another parental line can be used to increase statistical power for the experiment when the lines are not fixed for alternative alleles Assume a biallelic QTL segregating in the two parental lines Consider a hierarchical backcross design in which high and low ranking individuals selected in the parental lines are randomly mated to produce F which are again randomly , l mated to one of the unselected parental lines Two effects would occur in this scheme: 1) an increase in the frequency of heterozygous individuals in the F r which would increase the power to detect QTL; and 2) an increase in genetic variance of the backcross offspring due to linkage disequilibrium [3], which would decrease power to detect QTL 2.6.1 Frequency of heterozygotes ranking parents among offspring of high and low i Computation of the frequency of heterozygous F individuals was carried by using integrals simultaneously of the three normal distributions corresponding to the three genotypes at the QTL The selected proportion (p, ) for the high line is related to the proportion selected among individuals with genotypes QQ, Qq and qq (p poq and pqq) by the equation: , QQ out QQ, Qq or ) i f(xIG being the normal density given genotype (Gi Values p PQ and P can be found for a unique truncation point, qq) , QQ q, qq t, analytically In the examples considered in this paper Simpson’s rule [13] was used for increasing values of the abscissa until a value of t was found satisfying the above equation for a given p Computation of expected genotype s frequencies among high ranking individuals after selection in the high line was carried out by with = The same procedure was followed to obtain expected genotype frequencies using the among low ranking individuals in the low line same proportion selected, p as for the high line Computing the expected , s l frequency of heterozygous F offspring resulting from the cross of selected parents from each line was carried out with equation (3) of Gomez-Raya and Gibson [6] It assumes random mating and utilizes the frequencies among extreme individuals from each of the two parental lines L , L’ , Qq9 (pQ PQ p ) L 2.6.2 Increased variance among the l offspring of F individuals The effect of selection of high and low ranking individuals on genetic variance and on the estimation of heritability is discussed by Bulmer [4] and Gomez-Raya et al !7! Briefly, if selection of high and low ranking individuals is carried out in the same population then genetic variance increases in the selected group of parents QAS (1 + !s!)o’!; where QA is the genetic variance for the polygenic component in the unselected population, k, [z 4>(z)]/ps, z is the absolute value of standard normal deviates at cutoff points, 4>( z) is the ordinate and r is accuracy of evaluation The above equation can also be used when crossing high ranking individuals from one parental line with low ranking individuals from the other parental line as long as the parental populations have the same genetic variance and the same polygenic heritability and allowance is made for the different selection intensities for each genotype at the QTL and for each line For simplicity, an approximation to account for different selection pressures across genotypes was made by weighting the values of z and according to >(z) the proportion selected from each genotype and line in the computation of k = = by where zi&dquo; and 4>(z;&dquo;) are the absolute values of standard normal deviate and ordinate at truncation point for genotypei and line m (i had values 1, and for genotypes QQ, Qq and qq, respectively) An exact computation would be feasible by using all possible groups resulting in the F offspring and by including the variation of the means corresponding to l the different groups This variation is small, particularly for QTL of small effect, and the approach used accounts for the increased variance among offspring from selected parents l Following Bulmer (3!, the genetic variance in the F generation is (Fl) 0’ A (1/2)ksr2)u! The disequilibrium is halved in the next generation since l parents in the F are chosen randomly The genotypic variance among backcross l offspring from randomly mating F individuals to one of the parental lines (unselected) is: (1 + The heritability among backcross offspring is also increased and low ranking individuals in the parental lines: by selection of high Power calculation was carried out as described in the corresponding section but using the increased frequency of heterozygotes and heritability as computed above The estimates of the gene effect are biased if selection of parents is used Correction for selection bias in the estimates of the gene effect can be achieved after accounting for the increased variance among backcross offspring by * where a is the estimate of the gene effect from the experiment using selection, and a is the estimate of the gene effect after correction for selection bias Both * & and a are in phenotypic standard deviations units RESULTS Table I shows the gametic disequilibrium parameter for alternative number of alleles segregating in the parental populations The disequilibrium is high when alleles at each of the two loci are fixed in one population but they are rare in the other population If the alleles at the two loci are segregating at a similar frequency in the two populations then disequilibrium is small In this situation, the analysis could be performed within families allowing for different alleles at the marker to be associated with the same allele at the QTL in different families Maximum expected power and its standard deviation for three different crossing designs in a variety of situations (QTL effect, residual heritability, experiment size) for heterozygosity 1, 0.68 and 0.32, are given in tables II, 111 and IV, respectively The heritability including QTL and polygenic variation (max)) (h of the parental population having the highest possible value is given in these tables It can be observed that residual heritability has a small effect on power for the range of heritabilities in the parental populations Therefore, power figures can be considered as a good approximation when accurate estimates of heritability in the parental population are not available On the other hand, a reduction in the frequency of heterozygotes among F r individuals diminished the power for any situation considered For example, for residual heritability 0.2 and QTL effect 0.5 in an experiment with 200 piglets in a hierarchical backcross design, the maximum expected power with a significance level of 0.05 fell from 0.85 to 0.31 for heterozygosity and 0.32, respectively Power using hierarchical designs (backcross and intercross) increases in all cases with the exception of traits with high heritability when compared to traditional intercross designs (tables 77-7 V) Maximum expected power using hierarchical backcross designs is larger than using hierarchical intercross designs when the QTL effect is small The opposite occurs for large QTL effects The use of a larger experiment size increases power in all cases For example, the average maximum power across QTL size and residual heritability (table II) for heterozygosity in hierarchical backcross designs is 0.60, 0.79 and 0.91 for experiment sizes 200, 400 and 800, respectively Similarly, for hierarchical intercross designs, average maximum power is 0.58, 0.71 and 0.81 for experiment sizes of 200, 400 and 800, respectively Variation in power due to the sampling in the F should also be considl ered Hierarchical backcross designs show a much larger variation than either intercross designs (tables III and II! An example of how power, standard deviation of power and expected risk change for alternative mating designs is given in table V In this example, heterozygosity is 0.68, residual heritability of the trait is 0.2 and QTL effect is 0.5 phenotypic standard deviations with experiment size of 400 piglets The resulting expected power utilizing a hierarchical backcross design is maximum when using two boars and 20 sows (0.79) The corresponding standard deviation of power and expected risk are 0.27 and 10 %, respectively A compromise between expected power and risk can be taken For example, the use of five boars has a low expected risk of % with an expected power of 0.74 It is more obvious with a hierarchical intercross design where expected power using between one and ten boars is very similar (0.81 to 0.83) However, the expected risk is only low when using four or more boars Hierarchical and traditional intercross designs gave identical results at heritability 0.5 (tables 11-I! This is because traditional intercross is an extreme of hierarchical intercross (one sow per boar) which is the best allocation at high heritabilities Table VI shows the number of boars, expected power and risk for designs having maximum expected power and minimum risk in hierarchical intercross and backcross designs The residual heritabilities are 0.10, 0.20 and 0.50 and the QTL size ranges from 0.3 to 0.6 phenotypic standard deviations in an z experiment of 400 piglets with F heterozygosity of 0.68 It can be observed that the numbers of boars for maximum expected power and minimum risk are different in many instances but the values are not very far apart It should also be noticed that the optimal allocation of resources in the experiment varies with the QTL effect A small number of boars is required for small QTL effects which in turn would also have high risk case of resources Heterozygosity and power using phenotypic selection of extreme individuals in a hierarchical backcross design with 400 piglets, residual heritability 0.20 and varying percentage selected are depicted in figures and 2, respectively Parental lines were segregating at frequencies of 0.8 (selected to increase) and 0.2 (selected to decrease) A percentage selected of 100 is given for comparison with the unselected case Percentage selected of 0.5 % was the lowest attempted Heterozygosity increases with increasing selection pressure in each line and with the size of the QTL Following the same pattern, power increases with decreasing percentage selected being up to 21 % more (figure 2) The increase is small if the power in the unselected situation is close to or DISCUSSION Most previous research in QTL mapping has assumed that lines at crossing fixed for alternative alleles at the QTL [1, 2, 8] The availability of divergent inbred lines for traits of economical interest is a limiting factor in these studies are Domesticated breeds of pigs have been undergoing artificial selection to increase rate and litter size, for example Only for research purposes, selection criterion has been low growth rate or reduced litter size in experimental populations Therefore, it seems more reasonable to assume that the lines at crossing differ in allele frequency at the QTL rather than that they are fixed for alternative alleles The benefits of crossing lines are to increase gametic disequilibrium between marker and QTL alleles and to increase heterozygosity among F individuals l Gametic disequilibrium generated at crossing is high when the allele frequency at the two loci (either two markers or one marker and one QTL) is very different in the two parental lines Gametic disequilibrium between two linked markers could be estimated in chromosomal fragments by the use of the disequilibrium parameter proposed in this paper It would require that markers are not tightly linked so parental lines could be assumed to be in linkage equilibrium If the gametic disequilibrium is low (e.g Q < 0.2) then the analysis could be carried growth out within families as considered in this paper with only a small loss in power If the gametic disequilibrium is high (e.g S2 > 0.8) then a maximum likelihood approach could be developed allowing the same marker allele to be associated with the same QTL allele only in some families Consequently, power figures as given in this study represent a lower bound of the achievable power in those cases More work is needed to assess the gain in power in situations where disequilibrium parameter has intermediate values Power for QTL detection in experiments involving crosses between two outbred lines is higher than using within-line experiments when the frequency of the favourable allele is higher than 1/2 in one parental line and lower than 1/2 in another parental line The increased power can be attributed to the higher l frequency of heterozygous F individuals than in either parental breed The larger the difference in allele frequency between the two parental populations, the larger is the increase in power It was assumed in the computation of power that residual heritability was constant for varying QTL effects and for a given heterozygosity in the F l This represents a variety of situations in which parental populations can be segregating for a QTL at a different frequency and having a different heritability The approach taken in this paper was to compute also the heritability (comprising QTL and polygene variation) of the parental line with the highest l possible value for a given heterozygosity in the F At low residual heritability, the contribution of large QTL to the heritability is high For example, for residual heritability 0.1 and QTL effect of 0.6 phenotypic standard deviations, the heritability is 0.22 when the allele frequency in the parental population is 0.68 (table 111) In spite of the large contribution of the QTL variation to the heritability, changes in power for different heritability values are small Discrepancies in power comparing residual heritability 0.1 and 0.2 for the same QTL size range between 0.00 and 0.05 (table 111) Therefore, power figures, as given in this paper, are well approximated when precise estimates of heritabilities from parental population are not available The approximation is particularly good when the QTL has a small effect On the other hand, it was assumed when computing power that recombination between QTL and marker did not occur In most instances, recombination would occur If the recombination fraction between the marker and the QTL is c then the number of offspring should be increased in proportion to 1/( 1- 2c?to obtain the same power as with complete linkage !14! Power could be increased by using interval mapping instead of the use of a single marker !10! Most of the relevant information in interval mapping comes from the non-recombinant individuals The number of offspring required to obtain power as given in this paper should be increased in a proportion 1/(1-c’), where c’ is the recombination fraction between the two markers That is, around 25 % more offspring are needed to achieve power as given in this paper for interval mapping between two markers with a recombination fraction of 0.20 A larger experiment size will also be needed to achieve the same power figures as given in this paper when the marker is not fully informative The use of interval mapping can mitigate the lack of informative offspring by utilizing information corresponding to nearby informative flanking markers The models used in this paper ignore the possibility of a common full-sib family component such as may occur for some traits of economic interest in pigs Power for mapping QTL affecting those traits would be reduced !9! The approach used in this study to compute power could be modified to account for this problem by incorporating a term corresponding to the common family component in the standard error of the contrasts It has also been assumed that the heritabilities in parental populations as well as within the offspring of F remain unchanged The frequency of l heterozygotes at other loci affecting the trait could increase genetic variance l among F individuals and, therefore, heritability One generation of random mating is enough to restore Hardy-Weinberg equilibrium as would occur after l intercrossing F individuals However, changes in the genetic variance among offspring resulting from the hierarchical backcross design may occur If there are other QTL affecting the trait showing dominance or epistasis, then changes in the phenotypic variance and heritability could also occur Therefore, departures from the power as computed in this paper would be expected for traits that show heterosis such as reproductive traits For a given power, a backcross requires about twice as many progeny as an intercross design because the number of segregating meioses is doubled in the intercross [10] Experimental design for mapping QTL in pigs has the limitation that the number of piglets per litter has a maximum around ten This limits the size of the subgroups of progeny where the contrasts are carried out Hierarchical backcross and intercross designs can, however, be used in pigs and allow contrasts within boars with larger subgroups of progeny Power of intercross versus backcross depends on both heritability and QTL effect when using hierarchical designs At high heritability levels, the hierarchical intercross design is more powerful than the hierarchical backcross design The opposite occurs at low heritability values An interesting result was that power for small QTL effects was higher in a hierarchical backcross than in a hierarchical intercross The only difference between the two designs is that the hierarchical backcross ignores meioses from sows In a hierarchical intercross, contrasts between alternative alleles inherited from sows are small with a large variation given the small QTL effect and the small progeny group (ten piglets) However, the total number of degrees of freedom may be high because of the large number of sows On the contrary, in a hierarchical backcross each of the few boars has large subgroups of progeny which means that contrasts have low variation and hypothesis testing is carried out with a low number of degrees of freedom As a consequence of the above, power may be reduced in the intercross There is not a general experimental design which can allocate resources in an optimum way if the experiment size is fixed Power in experiments with a low number of boars is higher for small size of QTL and high heritability (table V7) The opposite occurs for QTL with large size and low heritability Risk measured as the probability of power lower than 0.5 due to sampling of F, boars is high with a low number of boars If the frequency of heterozygous l F individuals is low then the risk increases and the power decreases (results not shown) A compromise between expected power and risk should be made A possibility for experiments searching for QTL affecting traits with known l heritability is first to decide the potential QTL size and F heterozygosity detectable for a given experiment size (tables 77-7 V) Second is to choose a design for that QTL size with maximum expected power but conditional on an expected risk of less than a value of, say, 0.05 For example, for traits with low heritability and using hierarchical backcross designs with an experiment size of 400 piglets, the detectable QTL sizes are 0.3, 0.3 and 0.5 for a heterozygosity of 1, 0.68 and 0.32, respectively (tables 11 IV) Between four (maximum expected power) and eight boars (minimum risk) could be chosen for a heterozygosity of 0.68 and a QTL effect of 0.5 phenotypic standard deviations (table VI) If the heterozygosity is lower then the QTL will be very difficult to detect even using a design with optimal allocation of resources Dam and sire lines currently utilized for commercial production of pigs (e.g Large White, Landrace and Duroc, in Norway) can be considered as parental lines which are crossed to produce F individuals with high frequency of l heterozygotes Genotyping both boars (Fr ) and piglets (backcross to Landrace) and recording of traits in the slaughter pigs can be used to map QTL by contrasting inheritance of alternative alleles with large subgroups of progeny Utilizing both sire and dam lines would also be feasible in a similar fashion to a hierarchical intercross design However, the cross between F boars and l sows is not currently used by the industry The swine industry in other countries often uses four-way crosses For example, line A is crossed to line B to produce F line C is crossed to line D to produce F The cross l (AB), (CD) l (AB) (CD) l l F x F would also show increased heterozygosity and power with respect to experiments within line if the allele frequency is higher than 1/2 among F and lower than 1/2among F individuals and vice versa i (AB) (CD) l The use of commercial pigs for QTL mapping has two advantages: 1) saving in the cost of running the experiment since the carcasses can be sold at current prices in the market; and 2) findings can immediately be used in markerassisted selection in the commercial stocks Selective genotyping involving growing a large population but genotyping only those individuals whose phenotypes deviate far from the mean has been proposed to increase power for QTL detection [10, 11] In this paper an alternative use of selection to increase power is proposed Selection of extreme individuals in the parental lines segregating at a QTL can be used to increase l heterozygosity in the F and consequently, power of the experiment It has been shown that selection is more useful with experimental designs having intermediate values of power without selection The increase in power is up to 21 % A higher power would be feasible if high selection intensities are practised, for example, by screening large populations for extreme individuals The main restriction with either use of selection (extreme individuals in the parental lines or extreme progeny) is that in many instances, QTL mapping experiments are carried out for several traits, which makes impractical the use of selection ACKNOWLEDGEMENTS We are grateful for useful discussions We also thank an contributions This work has been supported Research Council (NFR), and the Norwegian Pig Breeders to Erling Fimland anonymous reviewer for by the Norwegian Association his/her (NORSVIN) REFERENCES Andersson L., Haley C.S., Ellegren H., Knott S.A., Johanson M., AndersK., Andersson-Eklund L., Edfors-Lilja 1., Fredholm M., Hanson 1., Håkanson J., Lundstr6m K., Genetic mapping of quantitative trait loci for growth and fatness in pigs, Science 263 (1994) 1771-1774 [2] Beckmann J.S., Soller M., Detection of linkage between marker loci and loci affecting quantitative traits in crosses between segregating populations, Theor Appl [1] son Genet 76 (1988) 228-236 [3] Bulmer M.G., The effect of selection on genetic variability, Am Nat 105 (1971) 201-211 [4] Bulmer M.G., The effect of selection on genetic variability: a simulation study, Genet Res 28 (1976) 101-117 [5] Geldermann H., Investigations on inheritance of quantitative characters in animals by gene markers Methods, Theor Appl Genet 46 (1975) 319-357 [6] Gomez-Raya L., Gibson J.P., Within-family selection at a specific locus in dairy cattle, Genome 36 (1993) 433-439 [7] Gomez-Raya L., Schaeffer L.R., Burnside E.B., Selection of sires to reduce sampling variance in the estimates of heritability by half-sib correlation, Theor Appl Genet 81 (1991) 624-628 [8] Haley C.S., Knott S.A., Elsen J.M., Mapping quantitative trait loci in crosses between outbred lines using least squares, Genetics 136 (1994) 1195-1207 [9] Knott S.A., Haley C.S., Maximum likelihood mapping of quantitative trait loci using full-sib families, Genetics 132 (1992) 1211-1222 [10] Lander E.S., Botstein D., Mapping mendelian factors underlying quantitative traits using RFLP linkage maps, Genetics 121 (1989) 185-199 [11] Lebowith R.J., Soller M., Beckmann J.S., Trait based analyses for the detection of linkage between marker loci and quantitative trait loci in crosses between inbred lines, Theor Appl Genet 73 (1987) 556-562 [12] SAS Language: Reference, Version 6, 1990, First Edition, Cary, NC, SAS Institute Inc., 1042 p [13] Smith W.A., Elementary Numerical Analysis, Prentice-Hall, New Jersey, 1986 [14] Weller J.L, Kashi Y., Soller M., Power of daughter and granddaughter designs for determining linkage between marker loci and quantitative trait loci in dairy cattle, J Dairy Sci 73 (1990) 2525-2537 APPENDIX 1: Heterozygosity in the cross of two outbred lines In this section, the conditions for increased heterozygosity in the cross between two segregating populations at a QTL is studied Consider two outbred lines B and C segregating at a biallelic QTL with a favourable allele frequency B p and pc for lines B and C, respectively Under Hardy-Weinberg equilibrium, the frequency of heterozygous individuals at the QTL in breed B is: Similarly, is the frequency of heterozygotes among l F individuals in the cross Rearranging the above equation where d = PB P the difference in allele frequency between the two breeds - c, The amount d(2p 1) is the excess of heterozygotes in the cross with B respect to line B The same reasoning can be used to obtain the frequency of : C heterozygotes in the cross as a function of d and p The following can be concluded by inspection of equations A1 and A2 c > i) If p 1/2 and PB > 1/2 then d (2 p 1) is positive and d (1 — p!) is B - negative ii) Ifpc is iii) is 1/2 and p B < 1/2 then B d (2 p 1) - is negative and d (1 - p ) C in the offspring of a cross is not line (B or C) with the highest heterozygosity parental c If p > 1/2 and p < 1/2 then d (2 p 1) is positive and B B - For i in the is < positive or positive iv) Ifpc positive ii, the heterozygosity < 1/2 and p > B 1/2 then B d (2 p 1) - is higher than d (1 - pc) positive and d (1 - p!) For iii or iv, the heterozygosity in the offspring of a cross is higher than the heterozygosity within either line, B or C Therefore, crossing between outbred lines segregating at a QTL with a frequency larger than 1/2 in one line and lower than 1/2 in the other leads to increased heterozygosity in F which increases statistical power for QTL , l mapping with respect to analysis within line APPENDIX 2: Contrasts in backcross and intercross designs Outbred lines B and C are segregating at a biallelic QTL (with alleles Q and q) with favourable allele (Q) at frequencies PB and p for lines B and C, c l respectively Offspring from a heterozygous (Qq) F boar inheriting allele Q can be QQ (with genetic value a) and Qq (with genetic value 6) Offspring inheriting q can be qQ (with genetic value 6) and qq (with genetic value -a) The genotype frequencies of each type of offspring depending on the mating of l F individuals with parental line B (backcross BC-B), parental line C (backcross l BC-C), or with other F individuals (intercross) are: Contrasts within boar for backcrosses with lines B and C are9 = a + b(1 - p and6 = cY + S(1 - , respectively Contrast for the intercross ) B )p C is a + (1 - PB Pc Therefore, for fully additive models (as assumed in -) ’ the paper) B a = = APPENDIX 3: Standard error of contrasts The following derivations are for a phenotypic variance of Consequently, other variances are expressed as a proportion of the phenotypic variance It is also assumed that there is no residual covariance between sires Consider first a hierarchical backcross design where y2!!! is the lth observation of phenotype on a piglet with marker allele k inherited from boar i mated to sow j The total variance is the variance of the means of the subgroups of progeny (mixed fulland half-sibs) being contrasted and has value where h2is the residual heritability, s is the number of sows per boar and p is the number of piglets per sow The total variance of the means is also V T V6+Vw, b where V and V are the components between and within families, respectively w The standard error of the contrast in a hierarchical cross design is two times the square root of V since the contrast is within the progeny groups inheriting &dquo; alternative alleles from their boar and half of the total number of daughters per boar would receive each allele The square of the standard error can be computed by: = since V (1/4)hr The same approach can be used for the traditional intercross case the total variance is given by = design In this where !2!! is the kth observation on pigletk with marker genotype j inherited in family i The standard error of the contrast in the traditional intercross design is Note that the traditional intercross design is a particular intercross design with one sow per boar Therefore, SE A4 when s in hierarchical intercross designs = case of was computed using a hierarchical ... Selection in parental lines to increase power for QTL mapping Selection of high ranking individuals in one parental line and of low ranking individuals in another parental line can be used to increase... J.M., Mapping quantitative trait loci in crosses between outbred lines using least squares, Genetics 136 (1994) 1195-1207 [9] Knott S.A., Haley C.S., Maximum likelihood mapping of quantitative trait. .. disequilibrium parameter has intermediate values Power for QTL detection in experiments involving crosses between two outbred lines is higher than using within-line experiments when the frequency